Lesson Explainer: Logarithmic Functions Mathematics

In this explainer, we will learn how to identify, write, and evaluate a logarithmic function as an inverse of the exponential function.

A logarithmic function is just the inverse of an exponential function. Before looking at logarithmic functions, however, letโ€™s consider a linear function, such as ๐‘“(๐‘ฅ)=3๐‘ฅโˆ’1, and its inverse. Recall that to find the functionโ€™s inverse, we would first rewrite it as ๐‘ฆ=3๐‘ฅโˆ’1. Then, we would exchange the variables ๐‘ฅ and ๐‘ฆ to get ๐‘ฅ=3๐‘ฆโˆ’1 and solve for ๐‘ฆ, giving us ๐‘ฆ=๐‘ฅ+13. Our calculations reveal that the inverse of ๐‘“(๐‘ฅ)=3๐‘ฅโˆ’1 is ๐‘“(๐‘ฅ)=๐‘ฅ+13๏Šฑ๏Šง. You can also define the inverse as ๐‘”(๐‘ฅ)=๐‘ฅ+13. Since ๐‘“(๐‘ฅ) and ๐‘”(๐‘ฅ) are inverses of each other, if the point (๐‘ฅ,๐‘ฆ) satisfies ๐‘“(๐‘ฅ), then the point (๐‘ฆ,๐‘ฅ) must satisfy ๐‘”(๐‘ฅ). For example, we can see that the point (1,2) satisfies ๐‘“(๐‘ฅ) because ๐‘“(1)=3(1)โˆ’1=2 and that the point (2,1) satisfies ๐‘”(๐‘ฅ), because ๐‘”(2)=2+13=33=1.

Examining the graphs of ๐‘ฆ=๐‘“(๐‘ฅ) and ๐‘ฆ=๐‘”(๐‘ฅ) for the two functions below, we can see that they are reflections of each other in the line ๐‘ฆ=๐‘ฅ.

Now letโ€™s consider the exponential function ๐‘“(๐‘ฅ)=5๏—. The inverse of this function is the logarithmic function ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Šซlog or ๐‘”(๐‘ฅ)=๐‘ฅlog๏Šซ. Suppose we are asked to find ๐‘“(1) for the exponential function ๐‘“(๐‘ฅ)=5๏—. We would substitute 1 for ๐‘ฅ to get ๐‘“(1)=5=5๏Šง. Next, suppose we are asked to find ๐‘”(5) for the logarithmic function ๐‘”(๐‘ฅ)=๐‘ฅlog๏Šซ. We would substitute 5 for ๐‘ฅ to get ๐‘”(5)=5log๏Šซ and ask ourselves the following question: โ€œWhat power is a base of 5 raised to in order to equal 5?โ€ Since the answer to the question is 1, we know that ๐‘”(5)=1. Notice that the point (1,5) satisfies the exponential function, while the point (5,1) satisfies the logarithmic function. Just as with the linear function and its inverse above, the coordinates of the points that satisfy the two functions are reversed, and the graphs of the two functions are reflections of each other in the line ๐‘ฆ=๐‘ฅ as shown.

This will be true for any base, ๐‘Ž, of an exponential function and its inverse logarithmic function.

Definition: Logarithmic Function

A logarithmic function is the inverse of an exponential function. For the exponential function ๐‘“(๐‘ฅ)=๐‘Ž๏—, its inverse logarithmic function is ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Œบlog or ๐‘”(๐‘ฅ)=๐‘ฅlog๏Œบ. If the point (๐‘ฅ,๐‘ฆ) satisfies the exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies the logarithmic function. That is, if ๐‘ฆ=๐‘Ž๏—, then ๐‘ฅ=๐‘ฆlog๏Œบ. The graphs of the two functions are reflections in the line ๐‘ฆ=๐‘ฅ.

Keep in mind that according to this definition, the exponential function ๐‘“(๐‘ฅ)=10๏— would have an inverse logarithmic function of ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Šง๏Šฆlog or ๐‘”(๐‘ฅ)=๐‘ฅlog๏Šง๏Šฆ. When the base is 10, however, by convention, there is no need to specify it in the logarithmic function. That is, we can just write ๐‘”(๐‘ฅ)=๐‘ฅlog so that log๐‘ฅ is taken to be log๏Šง๏Šฆ๐‘ฅ (which can be read as log base 10 of ๐‘ฅ or, simply, as log of ๐‘ฅ). Likewise, for the exponential function ๐‘“(๐‘ฅ)=๐‘’๏—, the inverse logarithmic function needs to be written in a special way. Instead of writing ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Œพlog or ๐‘”(๐‘ฅ)=๐‘ฅlog๏Œพ, we would write ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šงln or ๐‘”(๐‘ฅ)=๐‘ฅln (which can be read as the natural log of ๐‘ฅ).

Definition: Natural Logarithmic Function

A natural logarithmic function is the inverse of an exponential function with a base of ๐‘’. Given that ๐‘“(๐‘ฅ)=๐‘’๏—, its inverse natural logarithmic function is ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šงln or ๐‘”(๐‘ฅ)=๐‘ฅln.

Now letโ€™s look at some problems related to logarithmic functions.

Example 1: Finding the Inverse Logarithmic Function

The function ๐‘“(๐‘ฅ)=2๐‘’+3๏— has an inverse in the form ๐‘”(๐‘ฅ)=(๐‘Ž๐‘ฅ+๐‘)ln. What are the values of ๐‘Ž and ๐‘?

Answer

Recall that when finding the inverse of a linear function, we exchange the variables ๐‘ฅ and ๐‘ฆ and then solve for ๐‘ฆ. To find the inverse logarithmic function, we need to follow the same procedure. Letโ€™s begin by rewriting the given exponential function as ๐‘ฆ=2๐‘’+3๏—. After exchanging the variables ๐‘ฅ and ๐‘ฆ, we get ๐‘ฅ=2๐‘’+3๏˜. Subtracting 3 from both sides of the equation gives us ๐‘ฅโˆ’3=2๐‘’๏˜, and then dividing both sides by 2 gives ๐‘ฅโˆ’32=๐‘’.๏˜

Now, since a natural logarithmic function has a base of ๐‘’, letโ€™s take the natural log of both sides of the equation. After rewriting the equation as lnln๏€ผ๐‘ฅโˆ’32๏ˆ=๐‘’,๏˜ we can ask ourselves the following question to simplify the right side: โ€œWhat power is a base of ๐‘’ raised to in order to equal ๐‘’๏˜?โ€ The answer to the question is ๐‘ฆ, so the equation can be rewritten as ln๏€ผ๐‘ฅโˆ’32๏ˆ=๐‘ฆ or ๐‘ฆ=๏€ผ๐‘ฅโˆ’32๏ˆln. We can then replace ๐‘ฆ with ๐‘”(๐‘ฅ) to get ๐‘”(๐‘ฅ)=๏€ผ๐‘ฅโˆ’32๏ˆln and rewrite the function as ๐‘”(๐‘ฅ)=๏€ผ12๐‘ฅโˆ’32๏ˆln to put it in the form ๐‘”(๐‘ฅ)=(๐‘Ž๐‘ฅ+๐‘)ln. This shows that ๐‘Ž=12 and ๐‘=โˆ’32.

Note

Remember that if the point (๐‘ฅ,๐‘ฆ) satisfies an exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies its inverse logarithmic function. Letโ€™s find a point (๐‘ฅ,๐‘ฆ) that satisfies ๐‘“(๐‘ฅ)=2๐‘’+3๏— and check to see if the point (๐‘ฆ,๐‘ฅ) satisfies ๐‘”(๐‘ฅ)=๏€ผ12๐‘ฅโˆ’32๏ˆln. If (๐‘ฆ,๐‘ฅ) satisfies ๐‘”(๐‘ฅ)=๏€ผ12๐‘ฅโˆ’32๏ˆln, it will not prove that our answer is correct, but if (๐‘ฆ,๐‘ฅ) does not satisfy the function, we will know for certain that we made a mistake.

Since ๐‘“(1)=2๐‘’+3=2๐‘’+3๏Šง, the point (1,2๐‘’+3) satisfies ๐‘“(๐‘ฅ). This means that the point (2๐‘’+3,1) should satisfy ๐‘”(๐‘ฅ). We can determine if this is the case by finding ๐‘”(2๐‘’+3) as follows: ๐‘”(2๐‘’+3)=๏€ผ12(2๐‘’+3)โˆ’32๏ˆ=๏€ผ๐‘’+32โˆ’32๏ˆ=๐‘’=1.lnlnln

This shows that the point (2๐‘’+3,1) does, in fact, satisfy ๐‘”(๐‘ฅ), as it should.

In the next example, we will demonstrate the relationship between the domain and range of an exponential function and the domain and range of its inverse. Recall that if the point (๐‘ฅ,๐‘ฆ) satisfies an exponential function, then the point (๐‘ฆ,๐‘ฅ) satisfies its inverse logarithmic function. Thus, if ๐‘ฅ is an element of the domain of the exponential function, it is also an element of the range of the logarithmic function. Likewise, if ๐‘ฆ is an element of the range of the exponential function, then it is also an element of the domain of the logarithmic function. This is true for any point (๐‘ฅ,๐‘ฆ), so we know that the domain of the exponential function must be the same as the range of the logarithmic function. Likewise, the range of the exponential function must be the same as the domain of the logarithmic function.

Example 2: Finding the Domain of the Inverse of an Exponential Function

Consider the function ๐‘“(๐‘ฅ)=๐‘๏—, where ๐‘ is a positive real number not equal to 1. What is the domain of ๐‘“(๐‘ฅ)๏Šฑ๏Šง?

Answer

Recall that the domain of a function is the set of all possible input values, and the range of a function is the set of all possible output values. First, letโ€™s consider the domain and range of the function ๐‘“(๐‘ฅ)=๐‘๏—. Since the exponent in the functionโ€™s definition can be any negative value, any positive value, or 0, the functionโ€™s domain is all real numbers. We were given that ๐‘ is positive, so to help determine the functionโ€™s range, letโ€™s take a specific positive value to use as an example, ๐‘=2, which would give us the function ๐‘“(๐‘ฅ)=2๏—. A negative value of ๐‘ฅ, such as โˆ’3, gives ๐‘“(โˆ’3)=2=18๏Šฑ๏Šฉ; a positive value of ๐‘ฅ, such as 3, gives ๐‘“(3)=2=8๏Šฉ; and a value of 0 for ๐‘ฅ gives ๐‘“(0)=2=1๏Šฆ. Notice that the output value in each case is positive, so we know that the range of ๐‘“(๐‘ฅ)=๐‘๏— must be ๐‘“(๐‘ฅ)>0.

Since the exponent in ๐‘“(๐‘ฅ)=๐‘๏— is a variable, we also know that the function is an exponential function. Recall that the inverse of an exponential function is a logarithmic function. That is, if ๐‘“(๐‘ฅ)=๐‘๏—, then ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Œปlog. Also recall that the range of an exponential function is the domain of its inverse. In other words, the domain of the logarithmic function ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Œปlog must be ๐‘ฅ>0.

Note

We can verify our answer by again assuming that ๐‘=2 and by then graphing both ๐‘ฆ=๐‘“(๐‘ฅ) and ๐‘ฆ=๐‘“(๐‘ฅ)๏Šฑ๏Šง for the functions ๐‘“(๐‘ฅ)=2๏— and ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Šจlog as follows:

We can see that the graphs are reflections of each other in the line ๐‘ฆ=๐‘ฅ and that the graph of ๐‘ฆ=๐‘“(๐‘ฅ)๏Šฑ๏Šง is located in only the first and fourth quadrants. In other words, it has the ๐‘ฆ-axis as an asymptote and has only positive input values. This confirms that the domain of the function ๐‘“๏Šฑ๏Šง is, in fact, ๐‘ฅ>0.

Now letโ€™s look at how we can evaluate a logarithmic function.

Example 3: Evaluating a Logarithmic Function at a Given Point

Consider the function ๐‘“(๐‘ฅ)=(3๐‘ฅโˆ’1)log๏Šจ. If ๐‘“(๐‘Ž)=3, find the value of ๐‘Ž.

Answer

To find the value of ๐‘Ž, we can begin by substituting ๐‘Ž into the given function for ๐‘ฅ and 3 for ๐‘“(๐‘ฅ) to get 3=(3๐‘Žโˆ’1).log๏Šจ

Recall that a logarithmic function is the inverse of an exponential function and that if ๐‘ฆ=๐‘Ž๏—, then ๐‘ฅ=๐‘ฆlog๏Œบ. It follows that if ๐‘ฅ=๐‘ฆlog๏Œบ, then ๐‘ฆ=๐‘Ž๏—.

We can see that in this problem, the base ๐‘Ž is 2, the value of ๐‘ฅ is 3, and the value of ๐‘ฆ is 3๐‘Žโˆ’1. Substituting these values into the equation ๐‘ฆ=๐‘Ž๏— gives us 3๐‘Žโˆ’1=2๏Šฉ.

Simplifying gives 3๐‘Žโˆ’1=8, and solving for ๐‘Ž gives a solution of ๐‘Ž=3.

Note

By finding ๐‘“(3) for the function ๐‘“(๐‘ฅ)=(3๐‘ฅโˆ’1)log๏Šจ, we can check our answer. Substituting 3 for ๐‘ฅ gives us ๐‘“(3)=(3(3)โˆ’1)log๏Šจ. After multiplying 3 by 3, we get ๐‘“(3)=(9โˆ’1)log๏Šจ, and after subtracting 1 from 9, we get ๐‘“(3)=8log๏Šจ. To simplify the right side of this equation, we must ask ourselves the following question: โ€œWhat power is a base of 2 raised to in order to equal 8?โ€œ The answer is 3, so we know that ๐‘“(3)=3 and that our answer is correct.

In the next example, we will determine the base of a logarithmic function given a point that the functionโ€™s graph passes through.

Example 4: Completing a Function Using a Given Point

Given that the graph of the function ๐‘“(๐‘ฅ)=๐‘ฅlog๏Œบ passes through the point (1024,5), find the value of ๐‘Ž.

Answer

To find the value of ๐‘Ž, first we need to rewrite the logarithmic function ๐‘“(๐‘ฅ)=๐‘ฅlog๏Œบ as ๐‘ฆ=๐‘ฅlog๏Œบ. Since the graph of the function passes through the point (1024,5), we know that when ๐‘ฅ=1024, then ๐‘ฆ=5. This allows us to substitute these values into the function to get the equation 5=1024.log๏Œบ

We know that if ๐‘ฆ=๐‘ฅlog๏Œบ, then ๐‘ฅ=๐‘Ž๏˜, so it follows that 1024=๐‘Ž๏Šซ. One way to solve this equation for ๐‘Ž is to take the fifth root of each side as follows: 1024=๐‘Žโˆš1024=โˆš๐‘Ž4=๐‘Ž.๏Šซ๏Šซ๏Žค๏Žค

This shows that the value of ๐‘Ž is 4. However, without a calculator, it may be difficult for us to determine that the fifth root of 1โ€Žโ€‰โ€Ž024 is 4. One strategy we might employ to find the fifth root of 1โ€Žโ€‰โ€Ž024 is to recognize that 1โ€Žโ€‰โ€Ž024 is a power of 2. We can list the powers of 2 as follows: 2=22=2ร—2=42=2ร—2ร—2=82=2ร—2ร—2ร—2=162=2ร—2ร—2ร—2ร—2=322=2ร—2ร—2ร—2ร—2ร—2=642=2ร—2ร—2ร—2ร—2ร—2ร—2=1282=2ร—2ร—2ร—2ร—2ร—2ร—2ร—2=2562=2ร—2ร—2ร—2ร—2ร—2ร—2ร—2ร—2=5122=2ร—2ร—2ร—2ร—2ร—2ร—2ร—2ร—2ร—2=1024.๏Šง๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ

With this information, we can solve the equation 1024=๐‘Ž๏Šซ for ๐‘Ž by substituting for 1โ€Žโ€‰โ€Ž024 and grouping the 2โ€™s as shown: 1024=๐‘Žโˆš2ร—2ร—2ร—2ร—2ร—2ร—2ร—2ร—2ร—2=โˆš๐‘Žโˆš(2ร—2)ร—(2ร—2)ร—(2ร—2)ร—(2ร—2)ร—(2ร—2)=โˆš๐‘Ž2ร—2=โˆš๐‘Ž4=๐‘Ž.๏Šซ๏Šซ๏Šซ๏Šซ๏Žค๏Žค๏Žค๏Žค๏Žค

This method gives us the same value of 4 for ๐‘Ž that we got previously.

As a final example, letโ€™s look at a real-world problem.

Example 5: Solving a Real-World Problem Using Logarithmic Functions

The pH of a solution is given by the formula pHlog=โˆ’(๐‘Ž)H+, where ๐‘ŽH+ is the concentration of hydrogen ions. Determine the concentration of hydrogen ions in a solution whose pH is 8.4.

Answer

The concentration of hydrogen ions is represented by ๐‘ŽH+, so we must solve for this variable to answer the question. Since we are given that the pH of the solution is 8.4, we can begin by substituting 8.4 into the formula for the pH to get 8.4=โˆ’(๐‘Ž).logH+

After substituting, we can then multiply both sides of the equation by โˆ’1 to get โˆ’8.4=(๐‘Ž)logH+. Recall that when the base of a logarithm is not shown, it is assumed to be 10, so in order to help solve for ๐‘ŽH+, we can now rewrite the equation as โˆ’8.4=(๐‘Ž).log๏Šง๏ŠฆH+

We know that a logarithmic function is the inverse of an exponential function and that if ๐‘ฅ=๐‘ฆlog๏Œบ, then ๐‘ฆ=๐‘Ž๏—, so based on the information we have, we can write an exponential equation by substituting values or variables into ๐‘ฆ=๐‘Ž๏— for ๐‘Ž, ๐‘ฅ, and ๐‘ฆ. Since ๐‘Ž=10, ๐‘ฅ=โˆ’8.4, and ๐‘ฆ=๐‘ŽH+, we get the equation ๐‘Ž=10.H+๏Šฑ๏Šฎ๏Ž–๏Šช

This shows that the concentration of hydrogen ions in a solution whose pH is 8.4 is 10๏Šฑ๏Šฎ๏Ž–๏Šช.

Recall that the domain of a logarithmic function is ๐‘ฅ>0, so in this case, ๐‘ŽH+ must be a positive number. Its value is, in fact, positive because 10 raised to any power is greater than or equal to 0. The negative exponent in 10๏Šฑ๏Šฎ๏Ž–๏Šช only means that the value of ๐‘ŽH+ is less than 1. By using a calculator, we can see that its approximate value is 3.98ร—10๏Šฑ๏Šฏ or 0.00000000398.

Now letโ€™s finish by recapping some key points.

Key Points

  • A logarithmic function is the inverse of an exponential function.
  • For the exponential function ๐‘“(๐‘ฅ)=๐‘Ž๏—, its inverse logarithmic function is ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šง๏Œบlog or ๐‘”(๐‘ฅ)=๐‘ฅlog๏Œบ.
  • When the base of a logarithmic function is 10, there is no need to specify it. If ๐‘“(๐‘ฅ)=10๏—, then ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šงlog.
  • A natural logarithmic function is the inverse of an exponential function with a base of ๐‘’. If ๐‘“(๐‘ฅ)=๐‘’๏—, then ๐‘“(๐‘ฅ)=๐‘ฅ๏Šฑ๏Šงln.
  • The graphs of an exponential function and its inverse logarithmic function are reflections in the line ๐‘ฆ=๐‘ฅ.
  • The domain of a logarithmic function is ๐‘ฅ>0, which is the range of its inverse exponential function.

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